1

J.Opt.Soc.Am.A. 15, 326-339 (1998)

Reconstruction of the Point Spread Function of theHuman Eye From Two Double-Pass Retinal Images

Using Phase Retrieval Algoritms

Ignacio Iglesias, Norberto Lopez-Gil and Pablo Artal

Laboratorio de Optica, Departamento de Física,Universidad de Murcia, Campus de Espinardo (Edificio C), 30071 Murcia,

Spain. (http://lo.um.es)

In the double pass technique used to measure the optical performance of the eye, thedouble pass image is the crosscorrelation of the input spread function with the output spreadfunction [J.Opt.Soc.Am.A. 12, 195 (1995)]. Then, when entrance and exit pupil sizes are equal,the information on the point spread function is lost from the double pass image, although themodulation transfer function of the eye is obtained. A modification of the double passtechnique using unequal size entrance and exit pupil allows to record a low resolution version ofthe ocular point spread function [J.Opt.Soc.Am.A. 12, 2358 (1995)]. We propose in this paperthe combined use of these two double pass measurements as input in a phase retrievalprocedure to reconstruct the ocular point spread function. We use an adapted version of theiterative Fourier transform algorithm consisting of two steps, in the first one, error-reductioniterations with expanding weighting functions in the Fourier domain yield an estimation ofthe phase that serves as initial guess for the second step, consisting of cycles of hybrid input-output iterations. We first tested the robustness and limitations of the retrieval algorithmusing simulated data with and without noise. We also applied the procedure to reconstruct thepoint spread function from actual measurements of double pass retinal images in the livingeye.

1. INTRODUCTION

The double pass method is an objectivetechnique to estimate the optical performance ofthe human eye1-4. It is based in recording thelight reflected back in the retina when the eyeforms an image of an object test. This externalretinal image, usually called aerial or doublepass image, is used to calculate the ocularmodulation transfer function (MTF)5,6. Thedouble pass method in its conventionalconfiguration with equal size entrance and exitpupils only produces even aerial images7. It wasshown theoretically7 and in the human eye8,that the double pass image is related to theretinal image, the ocular point spread function(PSF), through a correlation operation, insteadof a convolution as was previously generally

assumed2-4. This implies that asymmetricaberrations, such as coma, are lost in the doublepass images, although the MTF is correctlycomputed from the double pass images. In arecent study8, we proposed a simplemodification of the double pass apparatus toovercome that limitation and to obtaininformation on the shape of the retinal image. Itconsists in the use of unequal entrance and exitpupil sizes, with one of them, usually theentrance pupil, being small enough (we used 1.5mm pupil diameter) to produce a retinal imagesimilar to a diffraction-limited pattern. Then, therecorded aerial image is the correlation of theradially symmetric near diffraction-limited inputspread function with the output spread functionthat wants to be measured. With this set-up, thesymmetry of the conventional double passconfiguration is broken and information on theactual shape of the point spread function is

2

revealed. However, since the cut-off frequencyfor a diffraction-limited system with a 1.5 mmdiameter pupil is 48 cycles/degree for 543 nmlight, the unequal pupil size double passtechnique produces a low resolution version ofthe PSF, only providing information on theoptical transfer function up to the spatialfrequency cut-off.

In this paper, we apply computationalphase retrieval techniques9 to extend the rangeof the optical transfer function (OTF) beyondthe cut-off frequency and to reconstruct theocular PSF. This reconstruction constitutes aphase retrieval problem, similar to others inAstronomy, Electron Microscopy or wavefrontsensing, where one wishes to recover phase fromintensity measurements. A widely usedapproach to solve this problem is the iterativeFourier transform algorithms10-12, with Fouriertransformation back and forth between theobject and Fourier domain when imposing themeasured data and additional constrains inboth domains.

Although it would be possible theapplication of the iterative Fourier transformalgorithms to only one intensity measurement,i.e., the autocorrelation of the PSF obtained withthe equal size pupil double pass set-up, thealgorithms always stagnate in non-correctsolutions. To overcome this problem in the PSFreconstruction, we have followed somehow thestrategy that Fienup and Kowalsky13 used forphase retrieval of a complex valued object: tointroduce as additional information to thealgorithm a low resolution version of the PSF,obtained with the unequal pupil size doublepass apparatus, besides the autocorrelation ofthe PSF. Moreover, we modified the iterativeFourier transform algorithm to be adapted to theparticular characteristics of our retrievalproblem. The proposed data processing consistsin two blocks of iterations: first, cycles of error-reduction iterations with expanding weightingfunctions in the Fourier domain13 to obtain anestimate of the phase, that serves as initial guessin the second block, consisting in cycles ofhybrid input-output iterations12 to finallyreconstruct the PSF.

Other procedures have been also used toestimate the point spread function in the humaneye. The aberroscope technique, both in thesubjective14 and objective15 versions, provideswith a polynomial expansion of the wavefront

aberration of the eye that can be used tocalculate the point spread function. TheHartmann-Shack sensor16 has also been used inthe eye to estimate the wave aberration andsubsequently the PSF. We present here adifferent approach to directly estimate theocular PSF from double pass images without theneed of intermediate data on the ocular waveaberration.

The organization of the paper is asfollows: after a general revision of the imageforming theory in the double pass technique, wetested the phase retrieval algorithm usingsimulated data with and without noise. Wedescribed a modified set-up to simultaneouslyrecord a pair of double pass images andpresented examples of application of the phaseretrieval technique to the measured double passretinal images.

2. IMAGE FORMATION IN THE DOUBLEPASS TECHNIQUE

Double pass images obtained with equal sizeentrance and exit pupils, iD(x), are always even-symmetric, and they are related to the retinalimage (ocular PSF) through an autocorrelationoperation7:

iD(x) = pD(x) ⊗ pD(−x) (1)

with pD(x) the PSF for a pupil of diameter D, xa two-dimensional spatial variable, and ⊗means convolution. The Fourier transform ofexpression [1] is:

ID(u) = MD(u)[ ]2

(2)

with MD(u) the MTF of the eye for a pupildiameter D and u a two-dimensional spatialfrequency variable. The ocular MTF can becomputed from the equal pupil sizes doublepass image, iD(x), although without informationon the phase transfer function (PTF). Adifferent situation occurs in the version of thedouble-pass technique using different entranceand exit pupil sizes8, with one of the pupilssmall enough to consider the eye near todiffraction limited. In this case, the resultingdouble pass image, id(x), is given by:

3

id(x) = pd(x) ⊗ pD(−x) (3)

with pd(x) being the near diffraction-limitedocular PSF for the small pupil with diameter d,that should be similar to an Airy pattern. TheFourier transformation of expression [3] is:

Id(u) = MD(u)Ad(u)exp −i(FD(u)[ ] (4)

with Ad(u) the Fourier transform of pd(x), whichis limited to the cut-off frequency, ud,corresponding to the small pupil with diameterd. If pd(x) is approximately a diffraction-limitedpattern, can be considered as radiallysymmetric, and from equation [4], the PTF isobtained in the spatial frequency interval [0,ud],by:

Fd(u) = tan−1 Im Id(u)[ ]

Re Id(u)[ ] (5)

While the equal pupil sizes double pass methodprovides the MTF, but is unable to obtain thePTF, the unequal pupil sizes double passtechnique allows to obtain information of both,the MTF and PTF but in a limited spatialfrequency range. To consider the eye diffraction-limited, the diameter (d) of the small pupil mustbe equal or smaller than 1.5 mm5, 17. Even forthis diameter, the retinal image could not bediffraction-limited, although its shape is radiallysymmetric when correctly centered17.Displacements of the artificial pupil withrespect to the natural pupil of the eye smallerthan 1 mm en each direction produces retinalimages that can be considered nearly radiallysymmetric.

3. RECONSTRUCTION OF THE PSF BYITERATIVE FOURIER TRANSFORMALGORITHMS

After establishing that the double pass imageswere the autocorrelation of the ocular PSF, itwas suggested7 the direct application of phaseretrieval algorithms, in particular the iterativeFourier transform algorithm, to decorrelate iD(x)and to obtain the ocular PSF from only onedouble pass measurement. However, theiterative Fourier transform algorithm requiresgood estimates of the object support, i.e., the set

of pixels in the image with intensity valuesdifferent than zero. In this particular case, wehad to estimate the support of pD(x) from itsautocorrelation. Although several strategies havebeen proposed18,19 to reconstruct the objectsupport from the autocorrelation support, itremains difficult to extract good estimations ofthe support of pD(x) only from measurements ofiD(x). In practical terms, this means that even inthe case of simulated data without noise, thereconstruction of pD(x) from its autocorrelation,iD(x), failed, yielding results containing twinimages, with partial reconstruction of pD(x) andpD(-x), both associated with the same MTF.

The solution we propose here is toreconstruct the PSF from two double passimages: its autocorrelation, iD(x), and a lowresolution version of the PSF, id(x), usingiterative Fourier transform algorithms. From theunequal pupil sizes double pass image, id(x), wedirectly calculate the PTF in the spatialfrequency range [0,ud], and an estimation of thesupport of pD(x). From iD(x), the MTF iscalculated in the complete spatial frequencyrange. The combined use of iD(x) and id(x) inthe algorithm restricts the problem to retrieve theinformation of the PTF in the region [ud,uD] ofthe spatial frequency domain. The phaseretrieval algorithm requires three functions(images) as inputs to reconstruct the PSF(pD(x)): the support of pD(x), a binary imagebeing one in the set of pixels over pD(x) isdifferent than zero; the MTF, MD(u), computedup to spatial frequency uD as the square root ofexpression [2]; and the PTF, Fd(u), obtained byexpression [5] up to the spatial frequency ud.The support of PSF is estimated from id(x) bythresholding it at an appropriate intensitylevel13. The support function is set one at thosepixels where id(x) is above the threshold valueand zero elsewhere. Since id(x) is theconvolution of pd(x) with a function similar toan Airy pattern, this estimation of the support ismore extended than the actual one.

We propose a two steps-method for thereconstruction of the PSF, or equivalently toestimate the complete PTF. The phaseestimation technique consists of cycles ofiterations of the Fourier transform algorithms12.

(a) (b)

(c)

Figure 1. (a) Flow chart of the iterative Fourier transform algorithm of error reduction (ER) or input-output (IO).The restrictions imposed in the Fourier and diffraction planes are computed from the double-pass images (see textfor details). (b) Flow chart of the iterative Fourier transform algorithm to obtain an initial estimation of the PTFfor the hybrid algorithm. (c) Schematic diagram of the complete method to reconstruct the PTF. Block 1 consistsin ER iterations with sequential phase retrieval and forced modulus to obtain an estimation of the PTF tha tserves as start for the second part of the algorithm (Block 2) consisting in the hybrid algorithm with the correctMTF.

The basics of this algorithm (figure 1 (a)) is to goback and forth between the object (PSF) planeand the Fourier (OTF) plane, imposing theconstraints on support and non-negativity in thePSF plane and the available information in theOTF domain. The Fourier transform of the kthestimation of the PSF, pkD(x) corresponds to thekth estimation of the OTF:

MDk (u)exp iFD

k (u)[ ] (6)

A new OTF is formed at this point using theknown MTF, MD(u), with the computed phasebeing changed by the PTF given by equation [5]in the spatial frequency range [0,ud]. The inverse

Fourier transform of the new OTF estimationyields p’ kD(x). A new function in the PSF planeis formed by:

p xp x x

xDk D

k+ ¢ œ

ŒÏÌÓ

1

0( ) =

if

( ) if gg

(7)

with g the set of points at which p’kD(x) violatesthe constraints in the object (PSF) plane, i.e., ifp’kD(x) is non-zero outside the support ornegative inside the support. This is the error-reduction (ER) version of the algorithm yieldingin each iteration a lower error in the estimationof the PSF. To follow the evolution of thealgorithm, we calculate in each iteration an error

F ud ( ) ( )M uD

support

FT FT-1

( )¢p xDk ( )p xD

k+1( )i xd

( )i xD

non negativity

OTF plane

PSF plane

OTF plane

PSF plane

Fd(u)

id(x)

support

FT

non negativity

FT-1

( )F uck+1

( )p xck+1

F uck++D1( )

C uc+D ( ) iD(x)

( )F uDk+1

( )¢p xck

MD(u)

stagnation? c=D?

yes yes

notnot

constant in [uc, uc+ D ]

Sequential phaseretrieval with forced

modulus

(ER)

Hybrid algorithm(ER+IO)

Initial estimation:constant phase PTF

BLOCK 1 BLOCK 2

5

parameter, a normalized root mean square error,in the Fourier plane by the expression:

M u M u M uDk

Du

Du

( ) ( ) ( )−∑ ∑2 2

(8)

where MkD(u) is the current modulus estimationbefore is substituted by MD(u) at each iteration.In order to overcome stagnation, usuallyappearing in the ER algorithm, we use the hybridalgorithm12 consisting in performing alternatecycles of ER and input-output (IO) iterations. Inthe input-output iterations, the step given byexpression [7] is changed by:

p xp x x

p x p x xDk D

k

Dk

Dk

+ ′ ∉− ′ ∈

1 ( ) = if

if

( )

( ) ( )

γβ γ

(9)

with β being a feedback parameter.A major difficulty for the reconstruction

of the phase (PTF) in the interval [ud, uD] withthe hybrid algorithm using a constant as initialestimation, is that the MTF, MD(u), has lowvalues for high spatial frequencies andincorporates little information to the algorithm inthat range. In consequence, the details in the PSFwith low intensity and high spatial frequency arenot correctly reconstructed. An efficientmodification in the data processing is to obtainan adequate initial estimation with the ERalgorithm using, instead of MD(u), its squareroot, to increase the dynamic range of the Fouriermodulus. In this part of the algorithm, we alsoincorporate a strategy13,20 of sequential phaseretrieval with weighted functions to improve theconvergence of the algorithm. A schematic flowchart of this part of the algorithm is shown infigure 1(b). The autocorrelation of a circle, usedas weighting function13, CC(u), is zero at spatialfrequencies larger than uc, which is a valueslightly larger than ud (the cut-off frequency forthe low resolution double pass image given byequation [3]). The constraints in the Fourierdomain are the weighted square root of the MTF

(

Mc(u) = MD(u)Cc(u) ) and the PTF in the[0,ud] interval. As initial guess of the PTF in theinterval [ud,uc], we use a constant phase. TheER algorithm iterates to estimate the phase in[ud,uc]. When the error value, calculated by [8],

with MC(u) instead of MD(u), does not decreasesignificatively after each iteration, the cut-offfrequency in the weighting function increases bya value ∆. The new initial guess for the phase isthe estimate obtained in the previous series ofiterations in the interval [ud,uC] and a constantagain for the rest of the function. This procedureis repeated by increasing the cut-off frequency inthe weighting function by steps of ∆ pixels toreach the spatial frequency limit uD .

The solution obtained at the end of thisprocedure of sequential retrieval with forcedmodulus (block 1 in figure 1 (c)) serves as theinitial estimate in a second block (block 2 infigure 1 (c)), where the hybrid algorithm (ER+IO)uses now the correct MTF as input.

4. COMPUTER SIMULATED RESULTS

We evaluated the reconstruction approachdescribed above with simulated data, before theapplication to the actual double passmeasurements in the eye. We first used noise-free data to establish the limit of the technique.Later, we added gaussian noise to the simulatedinput images to have a signal to noise ratiosimilar to that appearing in the measured doublepass retinal images. All the calculations wereperformed in a Silicon Graphics Power Challengeworkstation with four MIPS R8000 processors.The computer programs were written in C underthe KHOROS 2.0 software developmentenvironment for image processing21. We usedalong the whole calculation process 256x256pixels-double precision images. One iteration ofthe ER or IO algorithm takes approximately 0.6seconds to be completed and typically the wholeprocedure to estimate the PSF applying the twoblocks of figure 1 (c) takes around 30 minutes.Noise-free data

Let a simulated PSF, pD(x), normalizedto one and shown in figure 2 (a) as a contour lineimage. This PSF corresponds to a pure comaaberration (-1.47λ3cosθ) with (r,θ) coordinatesover the pupil with a diameter of 128 pixels(within a window of 256 pixels). The PSF issampled at exactly the Nyquist spatialfrequency and has a Strehl ratio of 0.17. Fromthis image, we calculated its autocorrelation,iD(x), shown in figure 2 (b), and the associatedMTF, MD(u), figure 2 (c) and PTF, FD(u)

Figure 2. Simulated data to test the reconstructionalgorithm. (a) PSF test, pD(x), computed from a purecoma aberration with a pupil diameter of 128 pixels(in a window of 256 pixels). (b) iD (x),autocorrelation of the PSF, pD(x). Both images arerepresented in a contour line graph subtending acentral region of 32x32 pixel extracted from the full256x256 pixel. (c) MTF, MD(u), represented in acontour map (256x256 pixels-image). (d) Principalvalue of the PTF, FD(u), represented in a grey levelimage. The cut-off frequency, uD,, is 128 pixels.

([0,uD=128 pixels]), figure 2 (d). The PSF for thesmall pupil diameter (similar to an Airypattern), pd(x), shown in figure 3 (a), wascalculated from the same coma aberration, butwith a pupil diameter of 48 pixels (within a 256pixels window) and has a Strehl ratio of 0.99.Figure 3 (b) shows id(x), and figure 3(c) thePTF,Fd(u), obtained from id(x), in a range of[0,ud=48 pixels]. The images of figures 2 (b)and 3 (b) are the available input data toreconstruct the PSF, simulating the pair ofmeasured double pass images. The support ofthe PSF is estimated by thresholding id(x), using0.01 as threshold value. From the pair of doublepass images (id(x), iD(x)), we obtain the PSFsupport, MD(u) and Fd(u) in [0,ud]; these are

Figure 3. (a) PSF, pd(x), for the small pupilcomputed from the same coma aberration as figure 2(a), but with a pupil diameter of 48 pixels (in awindow of 256 pixels). (b) id(x), convolution of pd(x)(panel (a) of this figure) and pD(x) (figure 2 (a)).,Both images are represented in a contour line graphsubtending a central region of 32x32 pixel extractedfrom the full 256x256 pixels. (c) Low spatialfrequency estimation of the PTF, Fd(u), computedfrom id(x). The cut-off frequency, uD. is 48 pixels.

the functions used as input in the reconstructionprocedure (figure 1 (a)).

Figure 4 presents results of three differentapplications of the procedure. These resultsjustify the final choice of the proceduredescribed in the previous section. First, we only

0.01

0.01

(a) (b)

0.01

(c) (d)

0.01

(a)0.01

(b)

(c)

7

used the hybrid algorithm (block 2 in figure 1(c))

Figure 4. Results obtained in the noise-freesimulation (see text for additional details). PSF (a )and associated PTF (b), obtained using only the block2 of the algorithm with a constant phase as initialguess. PSF (c) and associated PTF (d), obtained withonly the block 1 of the algorithm. This PTF is usingas initial guess for block 2 in the completereconstruction procedure. Finally reconstructed PSF(e) and associated PTF (f) obtained applying thecomplete algorithm.

with a constant phase as initial estimation. Weperformed 30 cycles, each one consisting of 10ER iterations followed of 90 IO iterations(β=0.7). Figure 4 shows the reconstructed PSF(a) and the principal value of the associatedPTF (b). The Strehl ratio for this PSF is 0.25 andthe reconstruction error calculated by expression[8] is 0.02. The normalized squared mean errorbetween the original PSF test (figure 2 (a)) and

the reconstructed PSF (figure 4 (a)) obtained bythe expression:

p x p x p xtD

fD

x

tD

x

( ) ( ) ( )−∑ ∑2 2 (10)

is 0.42. This is a large error between PSFs andthe reconstructed PTF (figure 4 (b)) isunderestimated by comparison with the originalPTF (figure 2 (d)). These results justify the needof obtaining an initial guess for the phase betterthan a constant. By using block 1, the expandingweighted forced Fourier modulus approach, toretrieve an estimation of the PTF over the wholespatial frequency range with ∆=2, we obtainedthe PTF of figure 4 (d). This is a better, lesserunderestimated, guess of the PTF. The objectassociated to this PTF and the correct MTF,MD(u), shown in figure 4 (c), reproduces mostdetails of the PSF test (figure 2(a)), althoughpresenting a higher Strehl ratio (0.27 versus 0.17in the PSF test). The error parameters obtainedby expressions [8] and [10] are 0.38 and 0.52respectively. The estimation of the phase offigure 4(c) was used as input in the block 2(ER+IO) of the complete procedure (figure 1 (c)).This block consisted in 30 cycles of 10-ERiterations plus 90-IO iterations with β=0.7.When the algorithm evolves, the MkD(u)estimations tend to MD(u), decreasing the Strehl

ratio of the pk+1D(x) estimations while thephase is approaching to the correct solution. Thefinal results of the PTF after the completeprocedure is presented in the figure 4 (f). Thelines of phase discontinuities are locatedapproximately in the same position than in thePTF test (figure 2 (d)). The final PSF, associatedto the PTF of figure 4 (f), is shown in figure 4(e).It reproduces the main spatial features of thePSF test (figure 2(a)), and its Strehl ratio is 0.18(to be compared with the 0.17 of the PSF test).The error parameters given by expressions [8]and [10] are 0.02, and 0.34 respectively. It mustbe noted that although the original and thereconstructed PSFs are visually very similar toeach other, there are a high error value betweenPSFs, due to the pixels of low intensity in theouter parts of the reconstructed PSF.Noisy data

We also tested the PSF reconstructionprocedure with noisy simulated double passimages as input. First, we evaluated the amountof noise that typically appears in the

0.01

(a) (b)

0.01

(c) (d)

0.01

(e) (f)

8

Figure 5. (a) Diffraction-limited MTF for a 4 mm pupil diameter (long dashed line). Radially averaged MTFsobtained from actual double pass images in two eyes (dashed line and dotted line). Radially averaged MTFcomputed from the simulated double pass image, iD(x) (figure 2(b)), contaminated with additive noise (solidline). (b) Radially averaged MTF computed from the simulated double pass image, iD(x) (figure 2(b)), withoutnoise (dotted line). Section of the adapted Butterworth filter (solid line). Radially averaged MTF computedfrom the simulated double pass image, iD(x) (figure 2(b)), contaminated with additive noise (long dashed line).Filtered MTF (dashed line) obtained from the noise contaminated iD(x).

experimental double pass images. Consideringthe exposure time and the average intensity onthe images, we assumed that the double passimages are contaminated with gaussian additivenoise. The effect of the noise contamination inthe double pass image, iD(x), on the MTF,calculated by the expression [2], is not the samefor different ranges of spatial frequencies. Whilethe MTF is practically unaffected at low spatialfrequencies and very little for intermediatespatial frequencies, the noise contaminationmainly affects the MTF at high spatialfrequencies. Typically, the MTF is mounted on apedestal of constant value at spatial frequencieslarger than the cut-off limit (uD). The value ofthe pedestal depends on the variance of thegaussian noise introduced in the double passimage, iD(x). To choose the value of the varianceof the noise to be added in the simulated doublepass images, we evaluated the pedestal valuesappearing in the MTFs computed fromexperimental double pass images. Figure 5 (a)shows two radially averaged MTFs (short-dashed lines) obtained from double pass images

for two subjects at 4 mm pupil diameter,compared with the diffraction-limited MTF forthe same pupil diameter (long-dashed line). Thesolid line in the same figure corresponds to theradially averaged MTF computed from thesimulated double pass image,iD(x), (figure 2 (b))after adding gaussian noise of zero mean andvariance 5*10-6. This noise produces a pedestalin the MTFs similar or slightly higher to thatfound in the MTFs obtained from actual doublepass images. Before using the MTF,MD(u), asinput for the algorithm, it is necessary toeliminate the pedestal produced by the noise. Ifthe pedestal is not removed, it will produce anoverestimated maximum peak in thereconstructed PSF that is not present in the PSFtest. To avoid this problem, we eliminate thehigh spatial frequency noise by multiplying theMTF by a two-dimensional Butterworth filter22of appropriate size. Figure 5 (b) shows anexample of the application of this filter to thesimulated data. The solid line is a section of the

0 20 40 60 80 100 120

0.0

0.2

0.4

0.6

0.8

1.0

Cycles/deg (pixel)

MT

F

0 20 40 60 80 100 120

0.0

0.2

0.4

0.6

0.8

1.0

Cycles/deg (pixel)

(a) (b)

9

Figure 6. Input data and results obtained in thesimulation with noise. (a) PTF, Fd(u), computedfrom the noise contaminated id(x), represented in agrey level image (256x256 pixels). (b) MTF, MD(u),(256x256 pixels) computed from the noisecontaminated iD(x). (c) reconstructed PSF (only thecentral 32x32 pixels of the full image is shown).

filter, the long-dashed line is the MTF obtaineddirectly from the noise-contaminated data, thedotted line is the MTF test (computed from thenoise-free PSF test) and the short-dashed line isthe contaminated MTF after being multiplied bythe filter. This operation keeps the correct valueof the MTF in most of the spatial frequencyrange and eliminates the problem of theoverestimated central peak in the PSF. On theother hand, the double pass image obtained with

unequal pupil diameters, id(x), does not requireany special pre-processing prior to be includedin the algorithm. It is only necessary to considerthat the low spatial frequency estimation of thePTF,Fd(u), appears contaminated at spatialfrequencies close to the cut-off frequency of thesmall pupil, ud.. Then, we will include Fd(u) inthe algorithm restricted to a spatial frequencylower than ud and then to reconstruct thecomplete PTF in a slightly larger region. Theresults of the simulation with noise arepresented in figure 6. Panel (a) is Fd(u)computed from the noisy id(x) and panel (b) isthe MTF, MD(u), computed from iD(x). The twodouble pass images were contaminated with thesame level of noise, using different seeds in thegeneration of the random numbers. The completecalculation procedure (the two steps of thealgorithm of figure 1 (c)) was applied to theseinput data. The reconstructed PSF is shown infigure 6 (c). The error parameters given byexpressions [8] and [10] are 0.16 and 0.34respectively, and the Strehl ratio 0.17, the sameas the PSF test (figure 2 (a)).

5. OCULAR PSF RECONSTRUCTIONFROM TWO DOUBLE PASSMEASUREMENTS

Once the retrieval technique was tested insimulated data with and without noise, this isapplied to actual measurements of retinalimages in two subjects. Since the two doublepass retinal images, id(x) and iD(x), have to beconsistent with the same optical conditions, toassure the convergence of the retrievalprocedure, we built a version of the double passapparatus to record both images simultaneously.This avoid the possible discrepancies betweenthe images if when recorded sequentially theconditions of centering or focus could changebetween the exposures.

Figure 7 shows a schematic diagram ofthe double pass setup that allows to recordsimultaneously two double pass retinal images.It is similar to that previously described8, butusing two recording paths separated by thebeam splitter BS2. In one path, the exit andentrance aperture diameter are the same (4 mm

(a)

0.11

(b)

0.01

(c)

10

in this case), while in the other path the exit

Figure 7. Schematic diagram of the double passapparatus to simultaneously record a pair of doublepass images. He-NE laser (543 nm); DF, neutraldensity filter; SF, spatial filter; M, mirror; L1-L3achromatic doublets; BS, BS1, BS2 pellicle beamsplitters. BS2 splits the output beam in two paths;one with a effective small pupil (1.5 mm diameter)and the other with a large pupil diameter (thesame as in the first passage). In the full frame of theCCD camera, two images are recorded: tha tcorresponding to the autocorrelation of the PSF,iD(x), and to the convolution of the PSF and the neardiffraction-limited pattern, id(x).

aperture has a 1.5 mm diameter. The CCDcamera (Spectrasource MCD1000) captures thetwo images with the same first passage, but witha different second passage. We use the fullframe of the CCD (512 square pixels), extractingeach individual image as 256x256 pixels, with16 bits per pixels. The sampling rate was 0.23minutes of arc per pixel. One double pass imageis the autocorrelation of the PSF, iD(x), when thedouble pass image is formed in the equal pupildiameters path. The other image, id(x), is a lowresolution version of the ocular point spreadfunction that corresponds to the convolution of

the point spread

Figure 8. Results in subject PA. (a) id(x), recordedwith 4 mm-1.5 mm pupil size configuration; (b)iD(x), recorded with 4 mm-4mm pupil sizeconfiguration. These two images are normalized toone and represented in a contour line plot. Only thecentral section of the full image is showed (64x64pixels corresponding to 14.7x14.7 minutes of arc). (c)PTF in the interval [0, ud=48 c/deg] obtained fromid(x) (panel (a)). (d) MTF (256x256 pixels,corresponding to 128 c/deg at the edge of the image)computed from iD(x). (e) Reconstructed PSF (64x64pixels). (f) Principal value of the retrieved PTF.

diffraction limited output image of the eye witha 1.5 mm diameter. Another possibility is torecord these two images sequentially afterchanging the aperture diameter in the first andsecond passages. If the experimental conditionsare correctly maintained, the computationalprocedure can be also applied in a similar

L2

L3

L1

BS 1SF

LED (950 nm)

Ne-HeLASER(543 NM)

P

CCD VIDEOCAMERA FOR

PUPILCENTERING

OC

SLOW SCANSCIENTIFICGRADE CCD

EXITPUPIL

(4 mm diameter)

EXIT PUPIL(1.5 mm diameter)

BS 2

BS

CAMERAOBJECTIVE

M

ENTRANCEPUPIL

(4 mm diameter)

LC DF

0.01

function with the near-

0.05

(a) (b)

0.11

(c) (d)

0.01

(e) (f)

11

0.04

0.01

(a) (b)

0.11

(c) (d)

0.01

(e) (f)

Figure 9. Results in subject NL. (a) id(x)., recordedwith 4 mm-1.5 mm pupil size configuration; (b)iD(x), recorded with 4 mm-4mm pupil sizeconfiguration. These two images are normalized toone and represented in a contour line plot. Only thecentral section of the full image is showed (64x64pixels corresponding to 14.7x14.7 minutes of arc). (c)PTF in the interval [0, ud=48 c/deg] obtained fromid(x) (panel (a)). (d) MTF (256x256 pixels,corresponding to 128 c/deg at the edge of the image)computed from iD(x). (e) Reconstructed PSF (64x64pixels). (f) Principal value of the retrieved PTF.

manner. The double pass retinal images werecollected in the right eye of two normal subjects(NL, male, 27 years old and 4 D myopic andPA, male, 35 years old and 2.5 D myopic) withthe apparatus described above. We recordedtwo double pass images: the equal size entranceand exit pupils and the unequal size entranceand exit pupils with monochromatic light (543

0.05

0.01

(a) (b)

0.04

0.01

(c) (d)

0.11

0.11

(e) (f)Figure 10. (a) Convolution of the reconstructed PSFfor subject PA with text diffraction- limited patternfor the 1.5 mm pupil diameter (64x64 pixels). (b)Autocorrelation of the reconstructed PSF for subjectPA (64x64 pixels). (c) Convolution of thereconstructed PSF for subject NL with thediffraction-limited pattern for the 1.5 mm pupildiameter (64x64 pixels). (d) Autocorrelation of thereconstructed PSF for subject NL (64x64 pixels). (e)and (f) MTFs computed from the reconstructed PSFsfor subjects PA and NL respectively. These imagesshould be compared with the experimental doublepass images and their associated MTFs of figures 8and 9 (see text for additional details).

nm), paralyzed accommodation, careful pupilcentering and the best refractive correction. Thepair of double pass images recorded in subjectPA are shown in figure 8. Panel 8 (a) shows, in acontour level plot, the central part (subtending14.7x14.7 minutes of arc) of the double passimage obtained with 1.5-4 mm pupil diameter,

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id(x); and in panel 8 (b) the double pass imageobtained with 4-4 mm pupil diameters, iD(x).The images corresponding to subject NL areshown in figures 9 (a) and (b). The PTF for bothsubjects (PA and NL) in the interval [0, ud=48c/deg] computed from id(x) are shown in figures8 (c) and 9 (c) respectively.

Pre-processing of the double-pass retinalimages

The calculation of the MTF from theequal pupil-size double pass image, iD(x),presents the typical problem of the DC peakappearing in the MTF due to the approximatelyconstant background of the double pass images.We faced before with that problem when weobtained the MTF from the double passimage6,8, either by subtracting a constant to theimage prior to compute the MTF or by removingthe DC peak directly from the MTF. A problemwith the subtraction procedure is how to chooseappropriately the value of the constant. Wepreviously used the average value in the fourcorners of the images and the mode of the image.However, sometimes even after subtraction ofthe constant value, a DC peak appears in theMTF. This means that either the double passimage is extending to the edge of the windowimage and then the DC peak is real or that thebackground is not constant through the image.The first possibility is not really happening withthe magnification used in our system. Thesimulated images shown in the previous sectionhave values different than zero in a small area(smaller than 64 pixels in the 256 pixels images).On the other hand, at very low spatialfrequencies, the MTF tends to be similar to thediffraction-limited MTF. However, when a noncorrect subtraction of the background isperformed, the MTF at low spatial frequenciesdrops dramatically presenting a different shapethan the diffraction-limited MTF. This impliesthat the subtraction of a constant value does notremove correctly the high values in the edges ofthe images, yielding a MTF inconsistent with theaberrations of the system. To solve this problem,we propose here to determine the support of thedouble pass image, iD(x), and to eliminate thepixels with values different than zero outsidethe support, prior to compute the MTF. Todetermine the support of the double pass image,we first apply a thresholding operation and latererosion and dilation operations22 to remove

possible isolated areas outside the support.Figures 8(d) and 9(d) presents the calculatedMTF for subjects PA and NL respectively afterperforming the above describe processing. Theradially averaged one-dimensional MTFs forthese results are in figure 5 (a). It must be notedthat both MTFs tend to the diffraction-limitedMTF at low spatial frequencies.

Reconstruction of the ocular PSFThe procedure to reconstruct the ocular

PSF was applied to the actual data, using thePTF in a region of [0, 32 c/deg], and the MTFobtained from the double pass image asdescribed above and multiplied by theButterworth filter. We used exactly the sameprocedure as in the simulations of the previoussection. The reconstructed PSFs are shown infigures 8 (e) and 9 (e) for subjects PA and NLrespectively. The Strehl ratios computed fromthese PSFs are 0.25 and 0.11. The finalnormalized error between MTFs (expression [8])were 0.05 and 0.08 for the two subjects. Theretrieved PTF in the whole range of spatialfrequencies are presented in figures 8 (f) and 9(f). They are only well defined to spatialfrequencies with relatively high values of themodulation (up to approximately 100 c/deg).

In figure 10, we show some results tofurther evaluate the validity of the reconstructedPSFs. Panels 10 (a) and (c) are the convolutionof the reconstructed PSF with the diffraction-limited patterns corresponding to a 1.5 mmpupil diameter. These images should becompared with the recorded double pass imagesof figures 8 (a) and 9 (a) respectively. Panels 10(b) and 10 (d) are the autocorrelation of thereconstructed PSFs for both subjects. Theseimage have to be compared with the originaldouble pass images of figures 8 (b) and 9 (b).Finally, the figures 10 (e) and (f) are the MTFsassociated to the reconstructed PSFs.

6. DISCUSSION AND SUMMARY

The conventional version of the double passtechnique, using equal entrance and exit pupildiameters, provides accurate estimates of theMTF but at the cost of loosing phaseinformation, that is the PTF and the actualshape of the PSF. A complete characterizationof the ocular optical performance should requireall that information. In many cases, the MTF issufficient, in particular when researchers are

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mainly interested in the relationship between theMTF and the psychophysical contrast sensitivityfunction. However, since odd aberrations arepresent in the eye, the determination of theactual retinal PSF is a significant advance. Froma fundamental perspective, it would contributeto better understand the image quality of theeye. From an applied point of view, the PSFresults would permit to develop new schematicmodels of the eye to serve as a reference in thedesign of ophthalmic devices. In addition, thedetermination of the wavefront aberration of theeye from double pass measurements by phaseretrieval techniques23,24 requires the actualestimates of the ocular PSF.

To extend the information available inthe double pass, we proposed8 a simplemodification of the technique to obtain a lowresolution retinal image, with information on thePTF in a limited range of spatial frequencies.Here, we combined this low resolution retinalimage, with the equal pupil sizes double passimage, to reconstruct the actual PSF of the eye,by using an iterative phase retrieval algorithm. Ithas been necessary to incorporate importantmodifications to the basic scheme of the iterativeFourier transform algorithm to assure a goodconvergence in the reconstruction of the ocularPSF. We have developed a two steps retrievaltechnique adapted to our particular problem. Inthe first step, cycles of error-reduction iterationswith a scheme of expanding weighted functionsin the Fourier domain, yields an estimation ofthe phase. That phase is used as initial guess ina second block of the algorithm consisting ofcycles of hybrid input-output iterations tofinally reconstruct the PSF. We first tested thevalidity and limitations of the algorithm withsimulated input data with and without noise.We presented reconstructed PSFs in the case oftwo eyes using as input in the computationalprocedure a pair of recorded double passimages. While with simulated noise-free data,we reconstruct the PSF with a normalized meansquare error in the MTF plane of about 0.02, andin the cases of actual double pass images, thefinal error in the reconstruction ranges from 0.05to 0.08. These reconstruction errors areacceptable in the case of results in human eye,where the collection of retinal images involveserrors, even with careful experimental conditionsof centering and focusing.

The reconstructed PSFs presented hereconstitutes, as a far as we know, the first

estimates of the retinal point spread functionobtained from double pass measurements in thehuman eye. As a possible direct applications ofthe results, we could mention the use of thereconstructed PSFs to digitally de-convolvefundus images to improve their contrast.

ACKNOWLEDGMENTS

This research was partially supported bya Spain DGICyT grant nº PB94-1138. Theauthors thank Javier Santamaría and PedroPrieto for critical revision of the manuscript.

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